The bite guards they sell in a sports store is designed to act as a cushion to protect your teeth in case of a blow to the face. The TMJ bite splint is not designed to act as a cushion but rather as a substitute for an even bite.
This is like when you have a wobbly table in a restaurant. If the waiter puts the correct size matchbook under the leg of the table it stops the wobble. If the matchbook isn’t exactly the right size, the table continues to wobble. So the best case is that you have a temporarily stable table. However, whenever the matchbook is removed, the legs are still not equal and the table will wobble again.
Similarly a bite splint will correct the wobble in your bite whenever you wear it, however like the matchbook, it must be exactly the right fit for it to work properly. And when it is removed your uneven bite is still there. Thus, a bite splint is not truly a long-term solution. The long term solution is to shorten the legs of the table until they are all equal such that you don’t have to use a matchbook at all. The equivalent of this is to even out the bite by equilibration.